Question

Evaluate without using trigonometric tables cot theta.tan(90-theta)-sec(90-theta)cosec theta + sin^2 65+ root 3( tan 5. tan 45. tan 85).
plz don't trick me.

Answer 1

Answer:

√3

Step-by-step explanation:

Note:  I think your question is not complete as you may have missed one more term i.e. + Sin²25. Without that you will not be able to evaluate without using trigonometric tables. Hence i am adding it as well in the solution below

cotθ.tan(90-θ) - sec(90-θ)cosecθ + sin²65+ Sin²25+ √3( tan 5. tan 45. tan 85).

Since Tan(90-θ) = Cotθ and Sec(90-θ) = Cosecθ, the above becomes

= Cotθ*Cotθ - Cosecθ*Cosecθ + Sin²65 + Sin²25 + √3( tan 5. tan 45. tan 85)

= Cot²θ - Cosec²θ + Sin²65 + Sin²25+ √3( tan 5. tan 45. tan 85)

Since Cosec²θ - Cot²θ = 1

= - 1 + Sin²65 + Sin²25 + √3 (Tan(90-85)*Tan45*Tan85)

Since Tan(90-θ) = Cotθ and Sin²θ - 1 = - Cos²θ.

=  - Cos²65 +  Sin²25+√3(Cot85° * 1 * Tan85°)

= √3 - Cos²65 + Sin²25

= √3 - [Cos (90-25)]² + Sin²25 (∵ Cos(90 - θ) = Sinθ)

= √3 - Sin²25 + Sin²25

= √3.